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V. K is closed under projective limits of inverse systems whose indexing sets are well-ordered and all whose mappings are onto. Examples of categories which satisfy these conditions will be given in Section 2. 1 ) The complete article is expected to appear in the Canadian Journal of Mathematics. Support from the National Research Council of Canada for the participation in this Symposium is gratefully acknowledged. B. BANASCHEWSKI 53 1. Projectivity. In the following, a mapping / : X -> Y between spaces will be called: a projection if f(X) = Y\ an essential projection if f(A) a X for any closed A c= X;2) a projective cover of Y (in a category K) if it is an essential projection such that there exist (in K) no non-trivial essential projections/' : X' -» X; a cross section of a mapping g : Y -> X if g o fis the identity mapping on X.

Theorem 1. For any separable metric space X and o-compact subset K of {X x s), (X x s) ~ (X x s) \ K. This theorem has some useful corollaries. Corollary 1. For any σ-compact subset K of s, s ~ s \ K. The proof of Theorem 1 is but a slight modification of a direct (independent) proof of Corollary 1. And Corollary 1 follows immediately from Lemmas 1 and 2. To obtain a second corollary of Theorem 1 we employ a theorem of Bessaga and Pelczynski ([4] or [6]) that (l2 x s) ~ l2. Corollary 2. For any σ-compact subset K of Z2, l2 ~ (¡2 N K)We are now in a position to outline a proof of the theorem that s ~ l2.

If every point of a space X has a neighborhood basis of power ^ m , and if Y is m-compact, then Y is upper compact with respect to X. Proposition 3. Let X be paracompact, and let Y be normal and upper compact with respect to X. Then X x Y is normal. 44 M. ATSUJI From this proposition and Proposition 2, we have the following corollary, which was proved by K. Morita [3], Th. 1. Corollary (Morita). Let X be a paracompact space such that each point has a neighborhood basis of power ^ m , and let Y be an m-compact normal space.

### 1934 General Motors GM Fisher Body Service Manual by General Motors

by George

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